Facing a problem in solving for $x$ from $-2x-1>49$

Question

Solve for $x$, $$-2x-1>49\tag1$$

$$-2x>49+1\tag2$$ $$x>50\div (-2)\tag3$$

$$x>-25\tag4$$

But my teacher says it is wrong!!

How? I have followed all the steps in my calculation!!

Answer 1

Let's take a different approach to see what went wrong. \begin{align*} -2x - 1 & > 49\\ -2x - 50 & > 0 && \text{subtract $49$ from each side of the inequality}\\ -50 & > 2x && \text{add $2x$ to both sides of the inequality}\\ -25 & > x && \text{divide both sides of the inequality by $2$} \end{align*} which can be written in the form $x < -25$.

Does this make sense?

Observe that $x = -30 < -25$. If $x = -30$, then $-2x - 1 = -2(-30) - 1 = 60 - 1 = 59 > 49$, so the inequality is satisfied.

On the other hand, $x = 0 > -25$. If $x = 0$, then $-2x - 1 = -2 \cdot 0 - 1 = 0 - 1 = -1 < 49$, so the inequality is not satisfied.

The rule that you need to keep in mind is that multiplying or dividing by a negative number reverses the direction of the inequality. For instance, $$2 < 3$$ but $$(-1)(2) = -2 > -3 = (-1)(3)$$

Let's modify your approach. \begin{align*} -2x - 1 & > 49\\ -2x & > 50 && \text{add $1$ to each side of the inequality}\\ x & < -25 && \text{divide both sides of the inequality by $-2$} \end{align*} where we have used the fact that dividing an inequality by a negative number reverses the direction of the inequality.

Answer 2

It is wrong because when you divided by $-2$ you should have reversed the inequality ($-2<0$).